To begin, we will record some facts about the set \(A\)  that were proved in [1]}. For comparison, the set \(A\)  in the statement of Proposition REF  corresponds to the set \(A^{\prime }\)  from [1]} (In [1]}, \(A^{\prime }\subset A\)  was selected to that \(\mathcal {E}_{\rho }(A^{\prime }+A^{\prime })\le K_3\mathcal {E}_{\rho }(A^{\prime })\)  for all \(\delta \le \rho \le 1\) . In Proposition REF , this property already holds for \(A\) ).
